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Let's have a look at how the matrix transpose is defined in the mathematical literature.  [Gantmacher, vol. 1, p. 230, 280,  Definition 10] defines the transpose to have the following properties; For a linear operator $A$ over $\mathbb F^n$, [check if general field F is implied] R^n$,  the transpose $A^T$ is the operator such that for any two vectors $x, y \in \mathbb R^n$,